## Thermodynamics B/C

UTF-8 U+6211 U+662F
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### Re: Thermodynamics B/C

What is the power required to isothermally compress 2 kg of a near-ideal helium gas from atmospheric pressure to five times atmospheric pressure in one minute
a) at 100 K
b) at 200 K
c) in terms of the temperature T

Justin72835
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### Re: Thermodynamics B/C

UTF-8 U+6211 U+662F wrote:What is the power required to isothermally compress 2 kg of a near-ideal helium gas from atmospheric pressure to five times atmospheric pressure in one minute
a) at 100 K
b) at 200 K
c) in terms of the temperature T
The molar mass of Helium is 4 grams/mol, so 2 kg equates to 500 moles. The work done during an isothermal compression is given by the following equation:

$W=nRT*ln(\frac{T_f}{T_i}=nRT*ln(\frac{P_i}{P_f})$

For A:

$500*R*100*ln(\frac{1}{5})=-669 kJ$

Dividing by 60 seconds gives a power of [b]-11.15 kW removed[/b].

For B:

$500*R*200*ln(\frac{1}{5})=-1338 kJ$

Dividing by 60 seconds gives a power of [b]-22.30 kW removed[/b].

For C:

$500*RT*ln(\frac{1}{5})=-6.69T$

Dividing by 60 seconds gives a power of [b]-111.5T W removed[/b].
"The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings."

University of Texas at Austin '23
Seven Lakes High School '19

UTF-8 U+6211 U+662F
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### Re: Thermodynamics B/C

Justin72835 wrote:
UTF-8 U+6211 U+662F wrote:What is the power required to isothermally compress 2 kg of a near-ideal helium gas from atmospheric pressure to five times atmospheric pressure in one minute
a) at 100 K
b) at 200 K
c) in terms of the temperature T
The molar mass of Helium is 4 grams/mol, so 2 kg equates to 500 moles. The work done during an isothermal compression is given by the following equation:

$W=nRT*ln(\frac{T_f}{T_i}=nRT*ln(\frac{P_i}{P_f})$

For A:

$500*R*100*ln(\frac{1}{5})=-669 kJ$

Dividing by 60 seconds gives a power of [b]-11.15 kW removed[/b].

For B:

$500*R*200*ln(\frac{1}{5})=-1338 kJ$

Dividing by 60 seconds gives a power of [b]-22.30 kW removed[/b].

For C:

$500*RT*ln(\frac{1}{5})=-6.69T$

Dividing by 60 seconds gives a power of [b]-111.5T W removed[/b].
Nice, your turn! (Also, there's a typo in the first line, should be Vf/Vi instead of Tf/Ti)

Note that the work done on the gas is -nRTln(Vf/Vi) though which makes it positive.

Justin72835
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### Re: Thermodynamics B/C

UTF-8 U+6211 U+662F wrote:
Justin72835 wrote:
UTF-8 U+6211 U+662F wrote:What is the power required to isothermally compress 2 kg of a near-ideal helium gas from atmospheric pressure to five times atmospheric pressure in one minute
a) at 100 K
b) at 200 K
c) in terms of the temperature T
The molar mass of Helium is 4 grams/mol, so 2 kg equates to 500 moles. The work done during an isothermal compression is given by the following equation:

$W=nRT*ln(\frac{T_f}{T_i}=nRT*ln(\frac{P_i}{P_f})$

For A:

$500*R*100*ln(\frac{1}{5})=-669 kJ$

Dividing by 60 seconds gives a power of [b]-11.15 kW removed[/b].

For B:

$500*R*200*ln(\frac{1}{5})=-1338 kJ$

Dividing by 60 seconds gives a power of [b]-22.30 kW removed[/b].

For C:

$500*RT*ln(\frac{1}{5})=-6.69T$

Dividing by 60 seconds gives a power of [b]-111.5T W removed[/b].
Nice, your turn! (Also, there's a typo in the first line, should be Vf/Vi instead of Tf/Ti)

Note that the work done on the gas is -nRTln(Vf/Vi) though which makes it positive.
You have a metal pot with heat capacity 540 J/K at a temperature of 114 °C. You then add 350 mL of water at 23 °C into the pot.

a. Is there any steam produced? (yes or no)

b. If yes, then what is the final temperature of the steam? If no, then what is the equilibrium temperature between the water and the pot?
"The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings."

University of Texas at Austin '23
Seven Lakes High School '19

UTF-8 U+6211 U+662F
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### Re: Thermodynamics B/C

Justin72835 wrote: You have a metal pot with heat capacity 540 J/K at a temperature of 114 °C. You then add 350 mL of water at 23 °C into the pot.

a. Is there any steam produced? (yes or no)

b. If yes, then what is the final temperature of the steam? If no, then what is the equilibrium temperature between the water and the pot?
Assuming the whole pot of water has to get to 100 degrees before any steam is produced:
Energy to heat water + energy to boil water + energy to heat steam = energy lost from metal pot.

=> T = -615 degrees Celsius, which is clearly impossible.

=> T = 47 degrees Celsius

a) No
b) 47 degrees Celsius

Justin72835
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### Re: Thermodynamics B/C

UTF-8 U+6211 U+662F wrote:
Justin72835 wrote: You have a metal pot with heat capacity 540 J/K at a temperature of 114 °C. You then add 350 mL of water at 23 °C into the pot.

a. Is there any steam produced? (yes or no)

b. If yes, then what is the final temperature of the steam? If no, then what is the equilibrium temperature between the water and the pot?
Assuming the whole pot of water has to get to 100 degrees before any steam is produced:
Energy to heat water + energy to boil water + energy to heat steam = energy lost from metal pot.

=> T = -615 degrees Celsius, which is clearly impossible.

=> T = 47 degrees Celsius

a) No
b) 47 degrees Celsius
"The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings."

University of Texas at Austin '23
Seven Lakes High School '19

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### Re: Thermodynamics B/C

Explain why we can make assumptions in the derivation of the ideal gas law, PV = nRT, such as "The number of molecules moving in each axis (x, y, and z) is equal".

Justin72835
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### Re: Thermodynamics B/C

UTF-8 U+6211 U+662F wrote:Explain why we can make assumptions in the derivation of the ideal gas law, PV = nRT, such as "The number of molecules moving in each axis (x, y, and z) is equal".
Not really sure about this one. My answer would be that because the molecules are so small, there are so many of them, and they have relatively little interaction with one another, you can assume that all the molecules follow Newton's Law of Motion and collide elastically with each other. From this, you are able to draw other conclusions as well.
"The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings."

University of Texas at Austin '23
Seven Lakes High School '19

UTF-8 U+6211 U+662F
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### Re: Thermodynamics B/C

Justin72835 wrote:
UTF-8 U+6211 U+662F wrote:Explain why we can make assumptions in the derivation of the ideal gas law, PV = nRT, such as "The number of molecules moving in each axis (x, y, and z) is equal".
Not really sure about this one. My answer would be that because the molecules are so small, there are so many of them, and they have relatively little interaction with one another, you can assume that all the molecules follow Newton's Law of Motion and collide elastically with each other. From this, you are able to draw other conclusions as well.
for the record, I was just looking for there being so many molecules that statistical treatment can be applied to them (especially that they have approximately random motion)

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### Re: Thermodynamics B/C

Two gases occupy two containers, A and B. The gas in A, of volume 0.14 cubic meters, exerts a pressure of 1.18 MPa. The gas in B, of volume 0.21 cubic meters, exerts a pressure of 0.82 MPa. The containers are united by a tube of negligible volume and the gases are allowed to intermingle. What is the final pressure in the container if the temperature remains constant?
"The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings."

University of Texas at Austin '23
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